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Homework Help: Isomorphism help

  1. Sep 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Is there an isomorphism from <R,+> to <R+,[tex]\times[/tex]> where [tex]\phi[/tex](r)=0.5[tex]^{r}[/tex] when r [tex]\in[/tex] R?

    2. Relevant equations
    For an isomorphism I know it is necessary to show there is a 1-1 and onto function. I am unsure if I can use the steps I am trying to use to show it is 1-1.

    3. The attempt at a solution

    For phi(r)=phi(s) I want to show r=s. Am I able to take the ln (or log?) of both sides to get ln(.05[tex]^{r}[/tex])=ln(0.5[tex]^{s}[/tex])? I am not sure which to use (ln or log) and where these logarithmic functions would be defined since for r=s, r and s are supposed to be real numbers.

    Thanks for the help.
     
  2. jcsd
  3. Sep 13, 2008 #2

    morphism

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    Maybe try rewriting 0.5^r=0.5^s as 2^r=2^s. This may make things cleaner.
     
  4. Sep 13, 2008 #3
    I don't know how to make that change?
     
  5. Sep 13, 2008 #4
    Even if that way does clean it up, is my way of taking ln of both sides wrong?
     
  6. Sep 13, 2008 #5

    Dick

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    Sure, just use logs. (0.5)^r=(0.5)^s iff r*log(0.5)=s*log(0.5). That shows it 1-1. Is it onto? But 1-1 and onto doesn't make it an isomorphism. You have to prove things like phi(r+s)=phi(r)*phi(s), right?
     
  7. Sep 14, 2008 #6
    Thanks for the help.
     
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